Finite Group Databasebeta

← C₇×C₂₃×C₆₇⋊C₃ #44569006

Subgroups — C₇×C₂₃×C₆₇⋊C₃ #44569006

Order 32361.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₂₃×C₆₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂₃×C₆₇⋊C₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂₃×C₆₇	C₃×C₇	1	1	yes	characteristic, complemented, Hall
4	C₇×C₆₇⋊C₃	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₇×C₂₃	—	67	1	no	maximal, Hall
6	C₇×C₆₇	C₃×C₂₃	1	1	yes	characteristic, complemented, Hall
7	C₆₇⋊C₃	C₇×C₂₃	1	1	yes	characteristic, direct product factor, Hall
8	C₇×C₂₃	C₆₇⋊C₃	1	1	yes	characteristic, direct product factor, Hall
9	C₃×C₂₃	—	67	1	no	Hall
10	C₆₇	C₃×C₇×C₂₃	1	1	yes	characteristic, complemented, Sylow
11	C₂₃	C₇×C₆₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
12	C₃×C₇	—	67	1	no	Hall
13	C₇	C₂₃×C₆₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
14	C₃	—	67	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇×C₂₃	C₆₇⋊C₃
commutator, cosocle, hypocenter	C₆₇	C₃×C₇×C₂₃
Frattini	C₁	G
Fitting, socle	C₇×C₂₃×C₆₇	C₃
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₇×C₂₃

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₇×C₂₃×C₆₇	C₃	semidirect
2. C₂₃×C₆₇⋊C₃	C₇	direct (N × H)
3. C₂₃×C₆₇	C₃×C₇	semidirect
4. C₇×C₆₇⋊C₃	C₂₃	direct (N × H)
6. C₇×C₆₇	C₃×C₂₃	semidirect
7. C₆₇⋊C₃	C₇×C₂₃	direct (N × H)
8. C₇×C₂₃	C₆₇⋊C₃	direct (N × H)
10. C₆₇	C₃×C₇×C₂₃	semidirect
11. C₂₃	C₇×C₆₇⋊C₃	direct (N × H)
13. C₇	C₂₃×C₆₇⋊C₃	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×C₂₃×C₆₇	3
C₂₃×C₆₇⋊C₃	7
C₇×C₆₇⋊C₃	23
C₃×C₇×C₂₃	67